Instead of repeating many steps similar to the case of an interval, we address new complications due to the change of geometry the remedies which resolve them. Through most of this section, we will use the existence of an optimal transport map; for all orders \(p\ge 1\)  there exists such a map \(T:S^1 \rightarrow S^1\) , as established for \(p>1\)  in see [1]} or [2]}, and for \(p=1\)  in [3]} (see [4]} and [5]} for the Eucledean settings, and [6]} for general metric settings.
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